Exact Optimal Solution of Fuzzy Critical Path Problems

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1 Available at Appl. Appl. Math. ISSN: Vol. 6, Issue (June 0) pp (Previously, Vol. 6, Issue, pp ) Applications and Applied Mathematics: An International Journal (AAM) Eact Optimal Solution of Fuzzy Critical Path Problems Amit Kumar and Parmpreet Kaur School of Mathematics and Computer Applications Thapar University Patiala-47004, India amit_rs_iitr@yahoo.com parmpreetsidhu@gmail.com Received: January 0, 0; Accepted: April 8, 0 Abstract In this paper, a fuzzy critical path problem is chosen to show that the results, obtained by using the eisting method [Liu, S.T.: Fuzzy activity times in critical path and project crashing problems. Cybernetics and Systems (), 6-7 (003)], could be improved to reflect, more appropriate real life situations. To obtain more accurate results of fuzzy critical path problems, a new method that modifies the eisting one is proposed here. To demonstrate the advantages of the proposed method it is used to solve a specific fuzzy critical path problem. Keywords: Fuzzy critical path problem, Linear Programming, Triangular fuzzy number MSC 000 No.: 03E7, 90C05, 90C70. Introduction In today's highly competitive business environment, project management's ability to schedule activities and monitor progress within strict cost, time and performance guidelines is becoming 5

2 5 Amit Kumar and Parmpreet Kaur increasingly important to obtain competitive priorities such as on-time delivery and customization. In many situations, projects can be complicated and challenging to manage. When the activity times in the project are deterministic and known, critical path method (CPM) has been demonstrated to be a useful tool in managing projects in an efficient manner to meet this challenge. The purpose of CPM is to identify critical activities on the critical path so that resources may be concentrated on these activities in order to reduce the project length time. The successful implementation of CPM requires the availability of clear determined time duration for each activity. However, in practical situations this requirement is usually hard to fulfill, since many of activities will be eecuted for the first time. To deal with such real life situations, Zadeh (965) introduced the concept of fuzzy set. Since there is always uncertainty about the time duration of activities in the network planning, due to which fuzzy critical path method (FCPM) was proposed since the late 970s. For finding the fuzzy critical path, several approaches are proposed over the past years. The first method called FPERT was proposed by Chanas and Kamburowski (98). They presented the project completion time in the form of fuzzy set in the time space. Gazdik (983) developed a fuzzy network of unknown project to estimate the activity durations and used fuzzy algebraic operators to calculate the duration of the project and its critical path. Kaufmann and Gupta (988) devoted a chapter of their book to the critical path method in which activity times are represented by triangular fuzzy numbers. McCahon and Lee (988) presented a new methodology to calculate the fuzzy completion project time. Nasution (994) proposed how to compute total floats and find critical paths in a project network. Yao and Lin (000) proposed a method for ranking fuzzy numbers without the need for any assumptions and have used both positive and negative values to define ordering which then is applied to CPM. Dubois et al. (003) etended the fuzzy arithmetic operational model to compute the latest starting time of each activity in a project network. Lin and Yao (003) introduced a fuzzy CPM based on statistical confidence-interval estimates and a signed distance ranking for (- ) fuzzy number levels. Liu (003) developed solution procedures for the critical path and the project crashing problems with fuzzy activity times in project planning. Liang and Han (004) presented an algorithm to perform fuzzy critical path analysis for project network problem. Zielinski (005) etended some results for interval numbers to the fuzzy case for determining the possibility distributions describing latest starting time for activities. Chen (007) proposed an approach based on the etension principle and linear programming (LP) formulation to critical path analysis in networks with fuzzy activity durations. Chen and Hsueh (008) presented a simple approach to solve the CPM problems with fuzzy activity times (being fuzzy numbers) on the basis of the linear programming formulation and the fuzzy number ranking method that are more realistic than crisp ones. Yakhchali and Ghodsypour (00) introduced the problems of determining possible values of earliest and latest starting times of an activity in networks with minimal time lags and imprecise durations that are represented by means of interval or fuzzy numbers. Shankar et al. (00) proposed a new approach for finding the total float of each activity, critical activities and critical path in a fuzzy project network. Kumar and Kaur (00) proposed a new method to find the fuzzy optimal solution of fully fuzzy critical path problems.

3 AAM: Intern. J., Vol. 6, Issue (June 0) [Previously, Vol. 6, Issue, pp ] 53 In this paper, a fuzzy critical path problem is chosen to show that the results, obtained by using the eisting method [Liu (003)], are not appropriate according to the real life situations. To obtain the appropriate results of fuzzy critical path problems, a new method is proposed by modifying the eisting method. To show the advantages of the proposed method over eisting method the chosen fuzzy critical path problem is solved by using the proposed method and it is shown that the appropriate results are obtained by using the proposed method. This paper is organized as follows: In Section, some basic definitions, an eisting approach for comparing fuzzy numbers and arithmetic operations between two triangular fuzzy numbers are presented. In Section 3, linear programming formulation of fuzzy critical path problems is presented. In Section 4, an eisting method for solving fuzzy critical path problems is presented. Shortcoming of an eisting method [Liu (003)] for solving a fuzzy critical path problem is discussed in Section 5. In Section 6, a method to find the maimum and minimum of two triangular fuzzy numbers is presented. In Section 7, by modifying an eisting method [Liu (003)], a new method is proposed to find the eact solution of fuzzy critical path problems. Advantages of the proposed method over eisting method are discussed in Section 8. In Section 9, the obtained results are discussed. Conclusion and future work is presented in Section 0.. Preliminaries In this section, some basic definitions, Yager's ranking approach for comparing fuzzy numbers and arithmetic operations between triangular fuzzy numbers are presented... Basic Definitions In this section, some basic definitions are presented [Kaufmann and Gupta (985)]. Definition. The characteristic function A of a crisp set A X assigns a value either 0 or to each member in X. This function can be generalized to a function A such that the value assigned to the element of the universal set X fall within a specified range i.e. : X [0,]. The assigned A value indicate the membership grade of the element in the set A. The function A is called the membership function and the set A = {(, ( )); X} defined by ( ) for each X is A A called a fuzzy set. Definition. A fuzzy number A = ( a, b, c) is said to be a triangular fuzzy number if its membership function is given by

4 54 Amit Kumar and Parmpreet Kaur 0, < a, ( a), a < b, ( b a) ( ) = ( c) A, b < c, ( b c) 0, c <. Definition 3. Two triangular fuzzy numbers A = a, b, ) and B = a, b, ) are said to be equal, i.e., A = B iff a = a, b = b, c = c. Definition 4. ( c ( c Let A = ( a, b, c) be a triangular fuzzy number and be a real number in the interval [0,] then the crisp set A = { X : ( ) } = [ a ( b a), c ( c b) ], is said to be -cut of A. A.. Yager's Ranking Approach Yager (98) proposed a procedure for ordering fuzzy sets in which a ranking inde (A) is calculated for the fuzzy number A = ( a, b, c) from its -cut A = [ a ( b a), c ( c b) ] according to the following formula: (A) = ( ( a ( b a) ) d + ( ( ) ) 0 c c b d ) = 0 a b c. 4 Let A and B be two fuzzy numbers, then (i) A B if (A) > (B), (ii) A B if (A) = (B), and (iii) A B if (A) (B). Since (A) is calculated from the etreme values of -cut of A i.e., a ( b a) and c ( c b), rather than its membership function, it is not required knowing the eplicit form of the membership functions of the fuzzy numbers to be ranked. That is, unlike most of the ranking

5 AAM: Intern. J., Vol. 6, Issue (June 0) [Previously, Vol. 6, Issue, pp ] 55 methods that require the knowledge the membership functions of all fuzzy numbers to be ranked, the Yager's ranking inde is still applicable even if the eplicit form the membership function of the fuzzy number is unknown. Remark.: Yager's ranking inde (98) satisfies the linearity property A ) = (A) + (B), R ( R is a set of real numbers). ( B.3. Arithmetic Operations In this section, addition and multiplication operations between two triangular fuzzy numbers are presented [Kaufmann and Gupta (985)]. Let A = ( a, b, c ) and A = ( a, b, c) be two triangular fuzzy numbers, then (i) A A = ( a a, b b, c c), (ii) A A ( a, b, c), where a = minimum ( aa, ac, ca, cc ), b = bb, c = ( aa, ac, ca, cc ), and maimum (iii) ( a, b, c ) A = ( c, b, a) Linear Programming Formulation of Fuzzy Critical Path Problems In this section, the fuzzy linear programming (FLP) formulation of fuzzy critical path problems is presented [Liu (003)]. Maimize t A A j, i) A ji = 0 i = i = n i N {, n} 0 ( i, A,

6 56 Amit Kumar and Parmpreet Kaur where, A : Set of all activities ( i,, : Fuzzy time duration of the activity ( i, j ), t N : Set of nodes, : Source node, n : Destination node. : time of the event occurring corresponding to the activity ( i,. 4. Eisting Method Liu (003) proposed a new method to solve the fuzzy critical path problems by representing all the fuzzy activity times as triangular fuzzy numbers. In this section, a brief review of an eisting method (Liu, 003) for solving the fuzzy critical path problems is presented. The steps of the eisting method are as follows: Step. Formulate the chosen fuzzy critical path problem into the FLP problem ( P ) : Maimize t A ji A j, i) A 0 0 ( i, A. Step. i = i = n i N {, n} = Convert the FLP problem ( P ) into the following crisp linear programming (CLP) problem ( P ) : Maimize ( t ) A ji A j, i) A 0 0 ( i, A. Step 3. i = i = n i N {, n} ( P ) = ( P )

7 AAM: Intern. J., Vol. 6, Issue (June 0) [Previously, Vol. 6, Issue, pp ] 57 Solve the CLP problem ( P ) to find the optimal solution { }. Step 4. Use the optimal solution { }, obtained in Step 3, to find the fuzzy critical path and also put the values of in t to find the maimum total fuzzy completion time of the project. A 5. Shortcomings of the Eisting Method In Step of the eisting method, to convert the FLP problem ( P ) into the CLP problem ( P ), only the ranking function is used i.e., to find the maimum of fuzzy numbers only rank of fuzzy numbers are compared but in literature [Kaufmann and Gupta (988)], it is pointed out that it is not possible to find the maimum value of different fuzzy numbers by using rank only and for this purpose in the literature mode and divergence are also used. In this section, a fuzzy critical path problem, chosen in Eample 5., is solved by using the eisting method and it is shown that the obtained results are not appropriate according to the real life situations. Eample 5.. The problem is to find the fuzzy critical path and maimum total fuzzy project completion time of the project, shown in Figure, in which the fuzzy time duration of each activity is represented by the following ( a, b, c) type triangular fuzzy numbers: t = (, 4, 6), t = (9,, 7), t =(7,9, ), t = (, 9, 6), and t = (6, 0, 4) (, 4, 6) (, 9, 6) (7, 9, ) 4 (9,, 7) (6, 0, 4) 3 Figure. Project network of the illustrated Eample 5.

8 58 Amit Kumar and Parmpreet Kaur Solution: The fuzzy critical path problem, chosen in Eample 5., can be solved by using the following steps of the eisting method [Liu (003)]. Step. Using Section 3, the fuzzy critical path problem, chosen in Eample 5., can be formulated as follows: Maimize ((, 4, 6) (9,, 7) (7, 9, ) (, 9, 6) (6,0, 4) ) + =, = 0, + = 0, + =,,,, 0. Step. Using ranking formula, presented in Section., the FLP problem, obtained in Step, can be written as: Maimize ( (,4,6) + (9,,7) + (7,9,) + (,9,6) + (6,0,4) ) + =, = 0, + = 0, + =,,,, 0, i.e., Maimize ( ) + =, = 0, + = 0, + =,,,, 0. Step 3. On solving CLP problem, obtained in Step, the following three optimal solutions are obtained: (i) = = and = = = 0, (ii) = = = and = = 0, (iii) = = and = = = 0.

9 AAM: Intern. J., Vol. 6, Issue (June 0) [Previously, Vol. 6, Issue, pp ] 59 Step 4. Using the values of, obtained from Step 3, the following three fuzzy critical paths are obtained : (i) 4, (ii) 3 4, (iii) 3 4. Putting the values of, obtained from Step 3, in [(, 4, 6) (9,, 7) (7, 9, ) (, 9, 6) (6, 0, 4) ], the maimum total fuzzy project completion times corresponding to paths 4, 3 4 and 3 4 are (4,, 3), (5,, 3) and (5,, 3), respectively. Let the obtained maimum total fuzzy project completion time (4,, 3), corresponding to path 4, is represented by T and the maimum total fuzzy project completion time (5,, 3), corresponding to both the paths 3 4 and 3 4, be represented by T, i.e., T = (4,, 3) and T = (5,, 3). If there eist more than one fuzzy critical paths for a project network problem then the maimum total fuzzy completion time of the project should be same corresponding to all the fuzzy critical paths but it can be seen from the obtained solution that the maimum total fuzzy project completion time corresponding to path 4 is different from the maimum total fuzzy project completion time corresponding to paths 3 4 and 3 4 i.e., T T, only T ) = T ) =. ( ( Since T and T are two different fuzzy numbers so their physical interpretation will also be different i.e., for the maimum total fuzzy completion time of a same project, two different interpretations will be required which may not be acceptable for real life problems. 6. Method for the Ordering of Two Triangular Fuzzy Numbers In this section, the eisting method [Kaufmann and Gupta (988)] to find order of two triangular fuzzy numbers, which will be used in the proposed method, is presented. Let A = ( a, b, c ) and B = ( a, b, c) be two triangular fuzzy numbers then use the following steps to compare A and B. Step. Find a b c ( A) = and 4 a b c ( B) =. 4 Case (i) If ( A ) > ( B), then A B, i.e.,

10 60 Amit Kumar and Parmpreet Kaur maimum { A, B} = A and minimum { A, B } = B. Case (ii) If ( A ) < ( B), then A B, i.e., maimum{ A, B} = B and minimum{ A, B } = A. Case (iii) If ( A ) = ( B) then go to Step. Step. Find mode ( A ) = b and mode ( B ) = b. Case (i) If mode (A) > mode (B) then A B, i.e., maimum{ A, B} = A and minimum { A, B } = B. Case (ii) If mode (A) < mode (B) then A B, i.e., maimum { A, B} = B and minimum{ A, B } = A. Case (iii) If mode (A) = mode (B) then go to Step 3. Step 3. Find divergence ( A ) = c a and divergence ( B ) = c a. Case (i) If divergence (A) > divergence (B) then A B, i.e., maimum{ A, B} = A and minimum{ A, B } = B. Case (ii) If divergence (A) < divergence (B) then A B, i.e., maimum{ A, B} = B and minimum{ A, B } = A. Case (iii) If divergence (A) = divergence (B) then A = B. 7. Proposed Method In this section, to overcome the shortcomings, discussed in Section 5, a new method is proposed to find the eact optimal solution of fuzzy critical path problems by modifying the eisting method (Liu, 003). The steps of the proposed method are as follow:

11 AAM: Intern. J., Vol. 6, Issue (June 0) [Previously, Vol. 6, Issue, pp ] 6 Step. Find the fuzzy critical path and maimum total fuzzy completion time of the chosen problem by using the eisting method, discussed in Section 5. There can be two cases: Case (i): If unique fuzzy critical path and hence unique fuzzy number, representing maimum total fuzzy project completion time, is obtained then the obtained maimum total fuzzy project completion time is the maimum total fuzzy completion time of the project and the obtained fuzzy critical path is the only fuzzy critical path of the project. Case (ii): If more than one fuzzy critical paths are obtained then go to Step. Step. Check that the fuzzy numbers, representing the maimum total fuzzy project completion time, corresponding to all the fuzzy critical paths are same or not. Case (i): If a unique triangular fuzzy number, representing maimum total fuzzy project completion time, is obtained then all the fuzzy critical paths, obtained in Step, corresponding to which the obtained fuzzy number was obtained, are fuzzy critical paths of the project and the obtained triangular fuzzy number will represent the maimum total fuzzy completion time of the project. Case (ii): If more than one triangular fuzzy numbers, representing maimum total fuzzy completion time of the project, are obtained then go to Step 3. Step 3. Let using the previous steps p different triangular fuzzy numbers T, T,..., T p, representing total fuzzy completion time, are obtained i.e., T i T j and ( T i ) = ( T j ) i j, i =,,..., p, j =,,..., p then find the optimal solution of the following CLP problem: Maimize ( mode ( a A, b, c )) ( a, b, c ) = ( Ti ), A ji = A j, i) A 0 0 ( i, A. i = or or... or p, i = i = n i N {, n}

12 6 Amit Kumar and Parmpreet Kaur Case (i): If by putting the obtained values of in ( ( a, b, c ) ) a unique triangular A fuzzy number, representing maimum total fuzzy project completion time, is obtained then the obtained triangular fuzzy number will represent the maimum total fuzzy completion time of the project and all the fuzzy critical paths may be obtained by using the obtained values of. Case (ii): If more than one triangular fuzzy numbers, representing maimum total fuzzy completion time of the project, are obtained then go to Step 4. Step 4. Let using the previous steps l triangular fuzzy numbers ' ' T, T,..., T ' l, where l p, representing maimum total fuzzy project completion time, are obtained, then find the optimal solution of the following CLP problem: Maimize ( divergence ( a A, b, c )) ( a, b, c ) = ( T ), A A mode ( a A, b j, i) A, c ji ) = 0 i ' = mode ( T ), i = or or... or p, i = i = n i N {, n} i i = or or... or l, 0 ( i, A. Now, putting the obtained values of in ( ( a, b, c ) ) A a unique triangular fuzzy number, representing maimum total fuzzy project completion time, will be obtained and the fuzzy critical path will be obtained by using the obtained values of. 8. Advantages of the Proposed Method Over the Eisting Method The main advantage of the proposed method over the eisting method is that on solving the fuzzy critical path problems by using the eisting method, the maimum total fuzzy completion time corresponding to two different fuzzy critical paths of a project may be different and due to which there will be different interpretations for the maimum total fuzzy project completion time of a same project which is not appropriate according to real life situations while by using the proposed method the maimum total fuzzy project completion time corresponding to all fuzzy

13 AAM: Intern. J., Vol. 6, Issue (June 0) [Previously, Vol. 6, Issue, pp ] 63 critical paths will be same. So, there will be a unique interpretation of maimum total fuzzy completion time of project. To show the advantages of the proposed method over eisting method the fuzzy critical path problem, chosen in Eample 5., for which the maimum total fuzzy project completion time corresponding to two different fuzzy critical paths are different, is solved by the proposed method and it is shown that by using the proposed method a unique fuzzy number, representing the maimum total fuzzy project completion time, is obtained corresponding to different fuzzy critical paths. 8.. Eact Optimal Solution of the Chosen Fuzzy Critical Path Problem The fuzzy critical path problem, chosen in Eample 5., can be obtained by using the following steps of the proposed method: Step. Using the results of Eample 5., the fuzzy critical paths for the chosen problem are 4, 3 4 and 3 4 respectively. Since more than one fuzzy critical paths are obtained, i.e., Case (ii) of Step of the proposed method is satisfied, so go to Step of the proposed method. Step. Using the results of the Eample 5., the maimum total fuzzy completion time of the project corresponding to the fuzzy critical paths 4, 3 4 and 3 4 are (4,, 3), (5,, 3) and (5,, 3) respectively. Since the fuzzy numbers (4,, 3) and (5,, 3), representing the maimum total fuzzy completion time of the project, are different, i.e., Case (ii) of Step of the proposed method is satisfied, so go to Step 3 of the proposed method. Step 3. Let the obtained maimum total fuzzy project completion time (4,, 3), corresponding to path 4, be represented by T and the maimum total fuzzy project completion time (5,, 3), corresponding to both the paths 3 4 and 3 4, be represented by T i.e., T = (4,, 3) and T = (5,, 3). Solving the CLP problem: Maimize ( mode (,4,6) + mode (9,,7) + mode (7,9,) + mode (,9,6) + mode (6,0,4) ) (,4,6) + (9,,7) + (7,9,) + (,9,6) + (6,0,4) =

14 64 Amit Kumar and Parmpreet Kaur (4,,3) or (5,,3), + =, = 0, + = 0, + =,,,, 0. The following three optimal solutions are obtained: (i) = = and = = = 0, (ii) = = = and = = 0, (iii) = = and = = = 0. Putting the obtained values of in ((, 4, 6) (9,, 7) (7, 9, ) (, 9, 6) (6, 0, 4) ), the obtained maimum total fuzzy project completion times are (4,, 3), (5,, 3) and (5,, 3) respectively. Since (4,, 3) (5,, 3) i.e., Case (ii) of Step 3 of the proposed method is satisfied, so go to Step 4 of the proposed method. Step 4. Let the obtained maimum total fuzzy project completion time (4,, 3), corresponding to ' path 4, be represented by T and the maimum total fuzzy project completion time (5,, 3), corresponding to both the paths 3 4 and 3 4, be represented ' ' ' by T, i.e., T = (4,, 3) and T = (5,, 3). Solving the CLP problem: Maimize ( divergence (,4,6) + divergence (9,,7) + divergence (7,9,) + divergence (,9,6) + divergence (6,0,4) ) (,4,6) + (9,,7) + (7,9,) + (,9,6) + (6,0,4) = (4,,3) or (5,,3), mode (,4,6) + mode (9,,7) + mode (7,9,) + mode (,9,6) + mode (6,0,4) = mode (4,,3) or mode (5,,3), + =, = 0, + = 0, + =,,,, 0

15 AAM: Intern. J., Vol. 6, Issue (June 0) [Previously, Vol. 6, Issue, pp ] 65 The optimal solution is = = and = = = 0. Putting the obtained values of in ((, 4, 6) (9,, 7) (7, 9, ) (, 9, 6) (6, 0, 4) ) a unique triangular fuzzy number (4,, 3), representing maimum total fuzzy project completion time, is obtained and using the same values of the obtained fuzzy critical path is Results and Discussion To show the advantage of the proposed over eisting method the results of fuzzy critical path problem, chosen in Eample 5., obtained by using the eisting and proposed methods is shown in Table. Eample 5. Table. Results of the eisting and proposed method Eisting method (Liu, 003) Fuzzy critical Maimum path total fuzzy completion time 3 4, 4 and 3 4 (5,, 3), (4,, 3) and (5,, 3) Proposed method Fuzzy critical path Maimum total fuzzy completion time 4 (4,, 3) It is obvious from the results shown in Table that the maimum total fuzzy project completion time, obtained by using the eisting method, is different corresponding to different fuzzy critical paths which is not appropriate according to real life situations, while on solving the same problem by using the proposed method a unique fuzzy number, representing the maimum total fuzzy project completion time is obtained. On the basis of these results it may be suggested that it is better to use the proposed method instead of eisting method for solving the fuzzy critical path problems. 0. Conclusion and Future Work By choosing some fuzzy critical path problems it is shown that it is not better to use the eisting method [Liu (003)] for solving the fuzzy critical path problems and a new method is proposed for solving the fuzzy critical path problems. To show the advantage of the proposed method over eisting method same fuzzy critical path problem is solved by using the eisting and the proposed method and it is shown that the results obtained by using the proposed method are better than the results obtained by using the eisting method. In the proposed method, the eisting method [Kaufmann and Gupta (988)] is used for the ordering of fuzzy numbers but since the eisting method [Kaufmann and Gupta (988)] is applicable only for the ordering of triangular fuzzy numbers. So the proposed method with eisting method [Kaufmann and Gupta (988)] can not be used for solving such fuzzy critical

16 66 Amit Kumar and Parmpreet Kaur path problems in which the fuzzy activity times are represented by other types of fuzzy numbers. In future the eisting method [Kaufmann and Gupta (988)] may be modified to find the eact ordering of other types of fuzzy numbers and then the proposed method with modified method can be used for finding the eact solution of such fuzzy critical path problems in which fuzzy activity times are represented by other types of fuzzy numbers. Acknowledgements The authors would like to thank the Editor-in-Chief "Professor Aliakbar Montazer Haghighi" and anonymous referees for various suggestions which have led to an improvement in both the quality and clarity of the paper. I, Dr. Amit Kumar, want to acknowledge the adolescent inner blessings of Mehar. I believe that Mehar is an angel for me and without Mehar's blessing it was not possible to think the idea proposed in this paper. Mehar is a lovely daughter of Parmpreet Kaur (Research Scholar under my supervision). REFERENCES Chanas, S. and Kamburowsk J. (98). The use of fuzzy variables in PERT, Fuzzy Sets and Systems, Vol. 5, No., pp. -9. Chen, S. P. (007). Analysis of critical paths in a project network with fuzzy activity times, European Journal of Operational Research, Vol. 83, No., pp Chen, S. P. and Hsueh, Y. J. (008). A simple approach to fuzzy critical path analysis in project networks, Applied Mathematical Modelling, Vol. 3, No. 7, pp Dubois, D., Fargier, H. and Galvagnon, V. (003). On latest starting times and floats in activity networks with ill-known durations, European Journal of Operational Research, Vol. 47, No., pp Dubois, D. and Prade, H. (980). Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York. Gazdik, I. (983). Fuzzy network planning- FNET, IEEE Transactions on Reliability, Vol. R-3, No. 3, pp Kaufmann, A. and Gupta, M. M. (985). Introduction to Fuzzy Arithmetics: Theory and Applications, Van Nostrand Reinhold, New York. Kaufmann, A. and Gupta, M. M. (988). Fuzzy Mathematical Models in Engineering and Management Science. Elsevier, Amsterdam. Kumar, A. and Kaur, P. (00). A new method for fuzzy critical path analysis in project networks with a new representation of triangular fuzzy numbers, Applications and Applied Mathematics: An International Journal, Vol. 5, No. 0, pp Liang, G. S. and Han, T. C. (004). Fuzzy critical path for project network, Information and Management Sciences, Vol. 5, No. 4, pp Lin, F. T. and Yao, J. S. (003). Fuzzy critical path method based on signed-distance ranking and statistical confidence-interval estimates, Journal of Supercomputing, Vol., No. 3, pp

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